Question: The sum of two numbers is $51$, and their difference is $17$. What are the two numbers?
Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 51}$ ${x-y = 17}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 68 $ $ x = \dfrac{68}{2} $ ${x = 34}$ Now that you know ${x = 34}$ , plug it back into $ {x+y = 51}$ to find $y$ ${(34)}{ + y = 51}$ ${y = 17}$ You can also plug ${x = 34}$ into $ {x-y = 17}$ and get the same answer for $y$ ${(34)}{ - y = 17}$ ${y = 17}$ Therefore, the larger number is $34$, and the smaller number is $17$.